 Inverse-square force

An inverse-square force is a central force whose magnitude varies as $1/r^2$, where $r$ is separation from the force centre. Both the gravitational force between point masses and the electrostatic Coulomb force between point charges have this dependence. The sign of the coupling determines whether the force is attractive or repulsive.

For an attractive force $\mathbf{F}=-k\mathbf{r}/r^3$ with $k>0$, a corresponding potential energy is $U=-k/r$, choosing zero at infinite separation. Moving twice as far from the centre reduces the force to one quarter, while halving the magnitude of this potential energy. The radial direction makes the torque about the centre vanish, so angular momentum is conserved. This reduces the orbital problem to planar motion and underlies the Kepler orbit.